#### What are the angle properties associated with circles?

The first fundamental property is the the radius of a circle meets a tangent at 90 degrees.

This result is self-evident, in that if the tangent were to be rotated clockwise or anticlockwise it would no longer make a right angle with the radius; the line would become a secant (a line that cuts a circle at two points), as opposed to a tangent (a line that strikes a circle at one point).

The second property is that the angle from any chord to the centre of the circle is twice the angle at the circumference.

This can be verified.

Clearly 2a + y = 180 and 2b + x = 180 (sum of angles in a triangle).

Also x + y + z = 360.

Therefore x + y + z = 2a + y + 2b + x.

Hence z = 2a + 2b = 2(a + b). That is, the angle at the centre is twice the angle at the circumference.

It should be immediately apparent that the position on the circumference is not important (excluding the alternate segment), hence we deduce a third property that from any chord, the angles on the circumference are equal.

By combining these results we are able to demonstrate the fourth property that the angle angle between a chord and a tangent is equal to the angle in the alternate segment.

This can be verified.

 We establish that x + y = 90 (radius meeting tangent) 2y + z = 180 (angles in a triangle) z = 2w (double centre angle)

As 2x + 2y = 180 = 2y + z, it follows that z = 2x = 2w.
Hence x = w.

As a result of the fourth property, we deduce the fifth property. In the special case where the chord is the diameter, the angle in the segment (semicircle) is 90 degrees.

This result can be verified directly.

Considering the triangle in the semicircle, a + a + b + b = 180.

Therefore 2a + 2b = 180, hence a + b = 90.

The sixth fundamental property relates to the cyclic-quadrilateral: opposite angles add to 180 degrees; that is, a + c = b + d = 180.

This can easily be proved by the second property.

As the angle at the centre is twice the angle at the circumference, we get
2a + 2c = 360, therefore a + c = 180.